Question

If the length of minor axis of an ellipse is \[ \frac13 \] times the sum of the distances of any point on the ellipse from its foci, then the eccentricity of the ellipse is

• A. \(\dfrac1{\sqrt2}\)
• B. \(\dfrac{2\sqrt2}{3}\)
• C. \(\dfrac{\sqrt2}{\sqrt3}\)
• D. \(\dfrac{2\sqrt2}{5}\)

Hint:

Remember the standard results: \[ \boxed{\text{Sum of distances from the foci}=2a,} \] \[ \boxed{\text{Minor axis}=2b,} \] and \[ \boxed{b^2=a^2(1-e^2).} \] These are sufficient to solve most ellipse problems involving eccentricity.

Answer 1

The Correct Option is B

Step 1: Use the given property of the ellipse. For an ellipse, \[ \text{Sum of distances from the foci} = 2a. \] Also, the length of the minor axis is \[ 2b. \] Given, \[ 2b=\frac13(2a). \] Hence, \[ b=\frac{a}{3}. \]

Step 2: Use the eccentricity relation. We know that \[ b^2=a^2(1-e^2). \] Substituting \[ b=\frac{a}{3}, \] we obtain \[ \frac{a^2}{9} = a^2(1-e^2). \] Therefore, \[ 1-e^2=\frac19, \] \[ e^2=\frac89. \]

Step 3: Find the eccentricity. Hence, \[ e = \sqrt{\frac89} = \frac{2\sqrt2}{3}. \] Therefore, \[ \boxed{\frac{2\sqrt2}{3}}. \] Thus, \[ \boxed{(B)} \] is the correct answer.